If you want to know more about Euclid and his influence, especially outside mathematics, read this. It is fascinating.
Originally posted on Boxing Pythagoras:
I have written, often, about one of my personal heroes from history, Euclid of Alexandria, who wrote a textbook called Elements which would serve as the foundation for all Western mathematics for 2000 years. You may recall that, outside of his name and a list of his writings, we know almost nothing about Euclid. We know nothing of his birth, or his schooling, or his politics. We don’t know if he traveled extensively or if he was relatively sedentary. We don’t know if he was tall, short, fat, skinny, handsome, or ugly. However, one thing we do know is that Euclid’s work, though purely mathematical, bore a tremendous influence on a wide variety of fields of knowledge.
Euclid’s Elements set out to prove the whole of mathematics deductively from very simple definitions, axioms, and postulates. Deductive logic provided a sound and absolute basis by which mathematics operated for every man…
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How to generate Pythagorean triples (example: 3,4,5), well one way at least.
Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy
we can write the difference of two squares
(x + y)2 – (x – y)2 = 4xy
and if we write x = A2 and y = B2 the right hand side is a square as well.
(A2 + B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2
which can be written as
(A2 + B2) 2 = (A2 – B2) 2 + (2AB) 2
the Pythagoras form.
Now just put in some integers for A and B
2 and 1 gives 3,4,5
Conjecture1: This process generates ALL the Pythagorean triples.
Conjecture2: Every odd number belongs to some Pythagorean triple.
My next post will be about finding the radius of the inscribed circle in a right angled triangle…..
Here’s my little story:
It was a class of day release students on a Higher National Certificate course in engineering. I reached a point in one class with a relationship between p and q, p = kq, with k a constant. “What’s its graph look like”, I asked. Deathly silence. “Ok, let’s try x and y”. Result y = kx. Same question, same response. “Well, what about y = 3x ?”. Same question, same response. So I wrote y = 3x + 2. Their eyes lit up, and they unanimously shouted “It’s a straight line!”.
The following is unreadable. Use the browser zoom or click the picture, or download the .doc file from https://howardat58.files.wordpress.com/2014/10/sums-of-powers.doc
In three hours from now (5.00pm PR time) takes place the last performance of Nana Badrena’s Ballet “DRACULA”, at the Teatro Yaguez in Mayaguez Puerto Rico. Created in 1999 in Cuba this ballet has been performed many times, in Cuba, Mexico, Pennsylvania, South America, and here. The ballet is based on the film from 1973 by Francis Ford Coppola.
This is our Ballet Company, Western Ballet Theatre, and we have a website http://www.wbtpr.com
(so it’s not just math !)(and theatre is the English spelling)
Dracula seduces Lucy Westenra
and then moves on to Mina, the reincarnation of his dead bride of 400 years earlier
The vampires in Dracula’s castle.
Jonathan, after Renfield grabs his valise
The dramatic bed scene, as the drugged Jonathan wakes up.
Filed under ballet, geometry
Found on the net. With avocados raining down on my porch roof this seems a good use for them!
“To hell with pumpkins, try the Guaco-lantern
So one day recently I was bored, and then the following rushed onto the page:
Half of a big pizza is equal to two small pizzas – rewrite this in as precise way as possible.
Is this 3 hours or 1/4 of a pizza?
How many hours equals 1/4 of a pizza?
Apologies to those for whom a clock face is a historical artifact.
How do I know it’s a pizza and not a cake?
Does half a day include half the night?
There are four feet in our yard, mine and my sister’s.
Would you prefer 1/2 of a round pizza or 1/2 of a square pizza?
Are ratios numbers or fractions (or neither) ?
Fractions are parts of the same whole. OK, I’ll have 5 quarters of that pizza (5/4 is a fraction, isn’t it?)
You cut, I choose !
This year fractions are parts of a whole. Next year fractions will be numbers. I guess the other party won the election.
2/3 and 4/6 are equivalent fractions. Equivalent to what ?
The word “fraction” has the same root as “fracture”. So something got broken. I think it was my faith in common sense.
The Common Core test question asked – How long is 3.25 hours. This could be 3 hours and 25 minutes or 3 hours and 15 minutes. I guess it depends on the grade level.
Back to the heavy stuff next time !
Filed under fractions, humor
I have seen some heavy handed ways of explaining the identity
a – (b + c) = a – b – c
Let us use algebra. Give the left hand side a name, say d .Then
a – (b + c) = d
This is an equation, so add (b+c) to each side and get
a = d + (b + c), then a = d + b + c as the parentheses are now superfluous.
Now subtract b from each side
a – b = d + c
Now subtract c from each side
a – b – c = d
so a – (b + c) = a – b – c
or is this too simple ? Look, no messing with p – q = p + -(q) stuff,
and no appeal to the famous distributive law.
You can do this, and other stuff, with numbers as well.
The derivative of the log function can be investigated informally, as log(x) is seen as the inverse of the exponential function, written here as exp(x). The exponential function appears naturally from numbers raised to varying powers, but formal definitions of the exponential function are difficult to achieve. For example, what exactly is the meaning of exp(pi) or exp(root(2)).
So we look at the log function:-